Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{3})^{-2}}}{{(a^{-3}t^{-1})^{-4}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{3}}$ to the exponent ${-2}$ . Now ${3 \times -2 = -6}$ , so ${(a^{3})^{-2} = a^{-6}}$ In the denominator, we can use the distributive property of exponents. ${(a^{-3}t^{-1})^{-4} = (a^{-3})^{-4}(t^{-1})^{-4}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{3})^{-2}}}{{(a^{-3}t^{-1})^{-4}}} = \dfrac{{a^{-6}}}{{a^{12}t^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-6}}}{{a^{12}t^{4}}} = \dfrac{{a^{-6}}}{{a^{12}}} \cdot \dfrac{{1}}{{t^{4}}} = a^{{-6} - {12}} \cdot t^{- {4}} = a^{-18}t^{-4}$.